Honors Calculus BC
Derivation of the Lateral Surface Area of a Frustum
The lateral area formula for a cone comes from the fact that the lateral area
of a cone is made of a circle where a wedge has been cut out to make it fit
around the base. It also helps to remember the definition of a radian – one
radian is the angle needed to make a piece of arc be the same length of the
s
radius of the circle, thus θ = ( where s is the arc length and l is the radius
l
of the circle, see the large circle below).
θ
π ⋅ l2
2π

θ 
= π ⋅ l 2 1− 
 2π 

s 
= π ⋅ l 2 1−

 2π ⋅ l 
 2π ⋅ l − 2π ⋅ r 
= π ⋅ l 2 1−



2π ⋅ l


 r
= π ⋅ l 2 1− 1− 
  l 
 r
= π ⋅ l 2 
 l
= π ⋅ l⋅ r
2
Cone Lateral Area
€ =π⋅l −
€
€
€
€
€
(
l
r
)
s
θ
l
r
€
The lateral€area formula for a frustum is based on the lateral area of the
cone, with the top removed. It takes advantage of the fact that the cone
removed is similar to the whole cone.
Frustum Lateral Area = π ⋅ r2 (s + l) − π ⋅ r1s
r1
= π (r2 s + r2 l − r1s)
l
= π ( r2 l + (r2 − r1 )s)
€
€
= π ( r2 l + r1l) (see below for this substitution)
€
= π ( r2 + r1 ) l
€
With our calculus calculation, as r2 → r1 → r
then = π ( r2 + r1 ) l goes to = 2π ⋅ rl .
€
€
r2
r1
r
= 2
s s+ l
r1 (s + l) = r2 (s)
r1 ⋅ l = r2 ⋅ s − r1 ⋅ s
r1 ⋅ l = s(r2 − r1 )
€
€
s
€